4. Multilayer Perceptrons

In this chapter, we will introduce your first truly deep network. The simplest deep networks are called multilayer perceptrons, and they consist of many layers of neurons each fully connected to those in the layer below (from which they receive input) and those above (which they, in turn, influence). When we train high-capacity models we run the risk of overfitting. Thus, we will need to provide your first rigorous introduction to the notions of overfitting, underfitting, and capacity control. To help you combat these problems, we will introduce regularization techniques such as dropout and weight decay. We will also discuss issues relating to numerical stability and parameter initialization that are key to successfully training deep networks. Throughout, we focus on applying models to real data, aiming to give the reader a firm grasp not just of the concepts but also of the practice of using deep networks. We put matters relating to the computational performance, scalability and efficiency of our models to subsequent chapters.

• 4.1. Multilayer Perceptrons
  • 4.1.1. Hidden Layers
  • 4.1.2. Activation Functions
  • 4.1.3. Summary
  • 4.1.4. Exercises
• 4.2. Implementation of Multilayer Perceptron from Scratch
  • 4.2.1. Initializing Model Parameters
  • 4.2.2. Activation Function
  • 4.2.3. The model
  • 4.2.4. The Loss Function
  • 4.2.5. Training
  • 4.2.6. Summary
  • 4.2.7. Exercises
• 4.3. Concise Implementation of Multilayer Perceptron
  • 4.3.1. The Model
  • 4.3.2. Exercises
• 4.4. Model Selection, Underfitting and Overfitting
  • 4.4.1. Training Error and Generalization Error
  • 4.4.2. Model Selection
  • 4.4.3. Underfitting or Overfitting?
  • 4.4.4. Polynomial Regression
  • 4.4.5. Summary
  • 4.4.6. Exercises
• 4.5. Weight Decay
  • 4.5.1. Squared Norm Regularization
  • 4.5.2. High-Dimensional Linear Regression
  • 4.5.3. Implementation from Scratch
  • 4.5.4. Concise Implementation
  • 4.5.5. Summary
  • 4.5.6. Exercises
• 4.6. Dropout
  • 4.6.1. Overfitting Revisited
  • 4.6.2. Robustness through Perturbations
  • 4.6.3. Dropout in Practice
  • 4.6.4. Implementation from Scratch
  • 4.6.5. Concise Implementation
  • 4.6.6. Summary
  • 4.6.7. Exercises
• 4.7. Forward Propagation, Backward Propagation, and Computational Graphs
  • 4.7.1. Forward Propagation
  • 4.7.2. Computational Graph of Forward Propagation
  • 4.7.3. Backpropagation
  • 4.7.4. Training a Model
  • 4.7.5. Summary
  • 4.7.6. Exercises
• 4.8. Numerical Stability and Initialization
  • 4.8.1. Vanishing and Exploding Gradients
  • 4.8.2. Parameter Initialization
  • 4.8.3. Summary
  • 4.8.4. Exercises
• 4.9. Considering the Environment
  • 4.9.1. Distribution Shift
  • 4.9.2. A Taxonomy of Learning Problems
  • 4.9.3. Fairness, Accountability, and Transparency in Machine Learning
  • 4.9.4. Summary
  • 4.9.5. Exercises